The Latin word “petitus” means inclining towards,
and the Latin word “fugo” means to drive away. Hence the European
scientists of the 17th century (who customarily composed their scholarly
treatise in Latin) used the terms centripetal and centrifugal to refer to
effects directed towards or away from (respectively) some central point. For
example, the effect of the Sun’s gravity was said to be centripetal because
it compels an orbiting planet toward the center of the orbit. On the other
hand, the inertia of an orbiting object is said to have a centrifugal effect,
because continued motion in a straight line would tend to carry the object
away from the center. In a circular orbit these two effects are equal, so the
object maintains a constant distance from the center.

The meanings of the words “centripetal” (inward) and “centrifugal”
(outward) are fairly clear and free of ambiguity, provided both the “central”
point and the location of the effect are adequately specified. However, these
words are often conjoined with the word “force”, the meaning of which has
been the subject of philosophical debate since ancient times. As a result,
the meanings of the terms “centripetal force” and (especially) “centrifugal
force” have sometimes been obscured.

According to Newton, a material object moves with constant
velocity unless acted upon by a force, in which case the object undergoes an
acceleration proportional to the applied force. More precisely, a system of
coordinates is defined such that the space coordinates of every free particle
are linear functions of the time coordinate. This leads to rectilinear
inertial coordinate systems. In terms of such coordinates, the applied force
is proportional to the second time derivative of the space coordinates. In
addition, Newton asserted that “to any action there is always an opposite and
equal reaction”. For example, the Sun acts on a planet by applying a
centripetal force that continually accelerates the planet, holding it in a
(roughly) circular orbit; and likewise the planet acts on the Sun by applying
a centripetal force that continually accelerates the Sun, holding it in a
(roughly) circular orbit. Newton was the first to recognize that the center
of the Sun is not actually the center of the orbits of the planets, because
the Sun itself orbits the true center of mass of the solar system. Of course,
the radius of the Sun’s orbit is extremely small compared with the radii of
the planetary orbits, but nevertheless the Sun does “orbit the Earth” in this
sense, just as the Earth orbits the Sun. (Eppur si muove!)

Notice that we have not mentioned any “centrifugal force”
in our description of orbiting bodies. This is consistent with the fact that
the only forces involved are the mutual forces of gravity that the Sun and
planet exert on each other, and these forces compel each body toward the
other, and therefore toward the center of their orbits. Now, one might argue
that there is some ambiguity in the direction of the forces based on the
ambiguity in the location of those forces. For example, the planet is
“pulling” the Sun inward (i.e., toward their common center of mass) by a
force acting in the direction from the Sun toward the planet. If we regard
this force as existing at the Sun’s location, then the indicated direction is
indeed “inward”, but if we regard this force as existing at the planet, then
the indicated direction of the force is actually “outward”. On that basis, we
might claim that the force of the planet on the Sun should be called centrifugal
rather than centripetal, and the same argument could be made for the force of
the Sun on the planet. This ambiguity is especially acute in the context of
the Newton-Cotes concept of gravity as an instantaneous “force at a distance”,
which makes the location of forces indeterminate. However, it is generally
agreed (in the Newtonian context) to identify a force with the associated
action, i.e., the deviation of an object from an inertial path. Thus the
force exerted by the planet on the Sun is regarded as being located at the
Sun, and therefore the force is properly called centripetal (as is the force
of the Sun on the planet).

There are, however, examples of genuine centrifugal
forces. For example, two electrons repel each other, so the forces could be
termed centrifugal, meaning the forces tend to drive the objects away from
the “center”. Of course, in such cases, the concept of a “center” is less
clear than in the case of closed orbital motion, but it still seems
legitimate to regard the source of the repulsion as the “center”.
Interestingly, Newton included a brief mention of this kind of “centrifugal
force” in the Principia. First he explained that the motion of an object
subject to a central force, whose magnitude varies inversely as the square of
the distance from the “central” point, is a conic section with the central
point at one focus. Then, in Book I, Section 3, Proposition 12, after proving
this for one branch of a hyperbolic path, he notes that “if this centripetal
force is turned into a centrifugal force, a body will move in the opposite
branch of the hyperbola”, as illustrated in the figure below.

Another situation in which centrifugal forces arise is in
cases exemplified by a marble rolling around the stationary circular housing
of a roulette “wheel”. This is similar to the case of a planet moving in a
circular orbit, except that the reaction force exerted by the marble on the housing
is applied at the point of contact, and hence is directed outwardly from that
point, as shown in the figure below.

Thus the marble exerts a genuine centrifugal (i.e.,
outward) force on the housing, balancing the centripetal (i.e., inward) force
exerted by the housing on the marble, so that the marble maintains a constant
distance from the center of the wheel. Indeed Newton used the term
“centrifugal force” to describe precisely this kind of force in Principia. (See
the Scholium following Proposition 4 of Section 2, Book 1.) However, it must
be remembered that Newton was writing in Latin, using the word centrifugal
simply as a literally descriptive adjective signifying the direction away
from the center.

Confusion about the meaning of the term “centrifugal
force” in modern English usage comes about because that Latin word has been adopted
to refer to something entirely different than the literal outward force
described above. Just to re-iterate, in the preceding example the housing
exerts a centripetal force on the marble, which causes the marble to undergo
centripetal acceleration, continually diverting it inward from its inertial
path, and compelling it to follow a circular path. This is the only force (in
the Newtonian sense of the word) being applied to the marble. Admittedly the
marble is, in turn, exerting a centrifugal force on the housing, but there is
no centrifugal (i.e., outward) force on the marble. The confusion
arises if we try to view the situation in terms of a system of coordinates
rotating (about the center of the roulette wheel) in such a way that the
marble is stationary. The housing is still exerting an inward force on the
marble and yet, in terms of this rotating coordinate system, the marble is
not accelerating. Needless to say, this is not a violation of Newton’s second
law, because that law – written in homogeneous form so that force is
proportional to the second derivative of the position coordinate with respect
to the time coordinate - applies only to motions described in terms of
inertial coordinate systems, whereas our rotating coordinate system is clearly
not an inertial coordinate system.

We might just choose to leave it at that, but in some
circumstances there is a desire to make use of Newton’s laws (formally) while
working in terms of a non-inertial coordinate system. This can actually be
done by introducing certain fictitious forces. For example, in the rotating
system of coordinates we must posit a centrifugal force on every (stationary)
particle, dependent on the rotational speed of the coordinate system, and
varying in proportion to the distance from the center of rotation. This
fictitious force exactly balances the inward force on the marble, so the
absence of acceleration (in terms of these rotating coordinates) is made
formally consistent with the homogeneous and isotropic form of Newton’s second
law. In other words, we explain why the marble is not accelerating by saying
that the net radial force on the marble is zero. Of course, in the inertial
sense, the marble actually is accelerating inward, but we are accounting
for one fiction by means of another. We are pretending, first, that the
marble is not accelerating, and second, that the marble is subject to an
outward (centrifugal) force – which explains why it is “not accelerating”.

In general we can consider a rigid Cartesian coordinate
system xyzt whose origin is co-located with the origin of some inertial
coordinate system XYZt, and whose absolute angular velocity is w(t). If an arbitrary vector q
is stationary in the rotating system, it follows that the derivative of q
with respect to time for the inertial system is given by the cross product

Now let r be an arbitrary vector, not necessarily
stationary in the rotating system, and let the scalars rx(t), ry(t),
rz(t) be the magnitudes of its components (as functions of time)
relative to the rotating system. Thus we have

where ux, uy, uz
are the unit vectors (also functions of time) of the xyz coordinate axes.
Differentiating each term with the chain rule and re-arranging the resulting
terms, we get the derivative of r with respect to t relative to the
inertial coordinates:

The expression in the first parentheses is simply the
derivative of the r vector in terms of the rotating coordinates. To
simplify the expression in the second parentheses, note that the unit vectors
are stationary in the rotating frame, so we can use the identity dq/dt
= wxq to write

Thus the derivative of r in terms of the inertial
XYZ coordinates is related to the derivative in terms of the rotating xyz
coordinates by

This applies to any vector. Hereafter for convenience we
will omit the subscripts and simply denote derivative in terms of the inertial
XYZ frame with ordinary “d” symbols, and derivative in terms of the rotating
xyz frame with “d” symbols.

Now, recall that Newton’s law of motion for a particle of
mass m is

To express this in terms of a reference frame rotating
with angular velocity w, we
need only apply equation (2) twice, as follows.

Expanding this expression, noting that the chain rule
applies to differentiation of cross products, and that the cross product is
distributive over addition, we have

The first term on the right hand side looks like the right
side of Newton’s law in an inertial frame, except that the differentiations
are in terms of the rotating coordinates, so the second derivative does not
represent the absolute acceleration. The second term on the right side is
sometimes called the Euler acceleration, and is proportional to the rate of
change of the rotating frame’s angular velocity. The above expression
indicates that this derivative is evaluated in terms of the rotating
coordinate system, but an equally valid application of formula (2) gives

Expanding and collecting terms, this gives

This is identical to equation (4), except that the
indicated derivative of w is taken with respect to the inertial coordinates.
It follows that

meaning that the vector corresponding to the rate of
change of the angular velocity of the rotating frame is the same, regardless
of whether the derivative is evaluated in terms of the inertial frame or the
rotating frame. On some level this might seem vaguely paradoxical, because one
might think that the rotating frame is always stationary with respect to
itself, by definition. Thus we have [w]xyz
= 0 at all times, and yet [dw/dt]xyz
need not be zero. Of course, there’s nothing surprising about the fact that a
function may be zero while its derivative is non-zero, but this condition
would ordinarily exist only for an instant, not for a continuous span of
time. This is a subtle but profound example of the difference between
inertial and non-inertial systems of reference.

The third term on the right side of (5) is commonly called
the Coriolis acceleration, and the fourth term is the centripetal (inward)
acceleration. Occasionally people find it convenient to bring some of the
acceleration terms over to the “force” side of the equation (with negated
signs) can treat them as fictitious forces. If all the accelerations are
brought over, the right hand side becomes zero, and we have dynamic
equilibrium, but it’s also common to leave the first term on the right hand
side, and just bring over the remaining terms. In that case Newton’s law is
written in the form

where

To illustrate, consider a reference frame rotating about
the z axis, and suppose a particle of mass m is moving with some velocity v
in the xy plane. Since each of the fictitious forces is given by a cross
product involving the angular velocity vector or its derivative, it follows
that they are all in the xy plane, except possibly for the Euler force in
case the axis of rotation is changing. The above equations show that the
Coriolis force is perpendicular to the velocity of the particle, with
magnitude 2mwr where r is the
distance from the origin. The centrifugal force points directly away from the
origin (i.e., the axis of rotation) with magnitude mw2r. Lastly, the Euler acceleration is
perpendicular to the radial direction, with magnitude mr(dw/dt), and lies in the xy plane if the axis
of rotation is fixed. This is illustrated in the figure below.

Notice that the centripetal acceleration term becomes a
centrifugal force when brought over to the force side of the equation. Newton
himself made use of the term “centrifugal force” in this fictitious sense.
The Scholium following Proposition 4 at the beginning of Book 3 explains the
reasoning by which Newton realized that the moon is held in its orbit around
the earth by the force of gravity, i.e., the same force that pulls terrestrial
objects (like apples) to the ground. He imagines several moons orbiting the
earth at different radii, and notes that Kepler’s law for orbiting bodies
implies that the inward acceleration is proportional to the inverse square of
the orbital radius. If we then imagine the lowest of these moons being at the
radius of the mountain tops on earth, we find that its downward acceleration
(in accord with Kepler’s law) has the very same value as the downward acceleration
of an apple at the top of the mountain. We must therefore conclude that the
inward force on orbiting bodies like the moon must be nothing other than the very
same force of gravity that pulls apples to the ground. Newton wrote

This
centripetal force [implied by Kepler’s law] would cause this little moon, if
it were deprived of all the motion with which it had remained in its orbit,
to descend to the earth – as a result of the absence of the centrifugal
force with which it had remained in its orbit – and to do so with the
same velocity with which heavy bodies fall on the tops of those mountains…

Here we see that Newton has tacitly asserted the existence
of a centrifugal (outward) force given to the “little moon” by its orbital
motion, and he conceives of this centrifugal force as balancing the
centripetal force, thereby maintaining the moon at its normal distance. This
illustrates how psychologically natural it is for us to “abstract away” the
acceleration of an object in circular motion and to conceive of a fictitious
(in Newtonian terms) outward force on the object to balance the real inward
force. (The same tendency can be seen underlying Galileo’s difficulty in
freeing himself from the idea that purely circular motion represented a kind
of force-free motion – the idea that prevented him from clearly articulating
the rectilinear law of inertia.)

Thus Newton uses the term “centrifugal force” in the
Principia to describe three very distinct concepts. First, he uses it
to refer to a hypothetical repulsive force (such as the force between two
electrons), which would result in a hyperbolic path, accelerating away from
the source of the “central” repulsive force. Second, he uses the term to
refer to the outward force exerted by a revolving object on some framework
(such as the force exerted by a roulette marble on the housing). Third, he
uses the term to refer to the “fictitious” outward force on a revolving
object when viewed from a revolving frame of reference. A fourth context in
which the concept of “centrifugal force” may arise is when phenomena are
described in terms of curved coordinate systems, such as polar coordinates.
Such non-linear coordinate systems are not inertial in the spatial sense,
even though they may be static (i.e., not accelerating), as discussed in the
note on Curved Coordinate Systems and
Fictitious Forces. A fifth usage of the term “centrifugal force” occurs
when the inertial forces on an object, relative to a momentarily co-moving
inertial frame, are de-composed into tangent and normal components (in the
osculating plane). The normal component is called centrifugal force. There is
no Coriolis force with this convention, because the particle is always at
rest with respect to the co-moving inertial coordinates. Needless to say, all
these usages are very closely related, and differ only by context and
convention.

Of course, if the origin of our rotating coordinate system
was undergoing translational acceleration, there would be an additional
acceleration term in equation (4), and that term could also be brought over
to the left side and treated as a fictitious force. Interestingly, a
fictitious force of that kind is found to behave exactly like a “real”
homogeneous gravitational field. This fact, due to the proportionality of
inertial and gravitational mass established by Galileo and Newton, served as
the inspiration for the Equivalence Principle, which led Einstein to the
general theory of relativity. According to that theory there is no local
physical difference between a “real gravitational force” and a “fictitious inertial force”, because free motion in a
gravitational field is understood to be purely inertial motion (locally). In
other words, the gravitational field and the inertial field are one and the
same, characterized by the ten metric tensor coefficients at each point of
spacetime. For more on this topic, see Vis
Inertiae.

The question of whether inertial forces are “real” or
merely “fictitious” has sometimes been passionately debated – as is usual for
matters of definition. One hears it stated confidently that fictitious forces
may be distinguished from real forces by the (alleged) fact that the latter
are mutually exerted between objects whereas the former – being supposedly
just an artifact of a choice of an accelerating coordinate system, are not.
However, strictly speaking, the assertion that inertia is intrinsic to each
body, rather than being a result of interactions with other objects in the
universe, is only a conjecture. Some scientists, notably Ernst Mach, have
maintained that inertia actually does arise from interactions with other
objects, albeit interactions of a kind different from those with which we are
most familiar. Indeed Einstein's general theory of relativity provides some
(limited) support for this view, since the inertial behavior of each object
is affected by the presence of other objects. Whether it is possible to
account for all inertia in this way is an open question, and depends on
subtle issues of boundary conditions and the topology of the universe. (A
prominent advocate of this view was the late American physicist John
Wheeler.) In the context of the "standard model" of quantum field
theory, there have been intense efforts to detect the so-called Higgs
particle, which according to the standard model of particle physics is an
excitation of a field (the Higgs field) responsible for the inertial masses
of most other elementary particles. The Large Hardon Collider (LHC) has
reported that the Higgs particle has indeed been detected. Nevertheless,
the Higgs mechanism does not explain or provide a mechanism for inertia
itself, it “merely” represents the mechanism whereby energetic fields (which
already have inertia proportional to their energy) acquire a rest frame with
a speed below the speed of light, and hence some of their energy takes on the
form of rest mass. The Higgs mechanism does not account for why energy has
inertia in the first place. Thus, we still have no definitive theory of the
origin of inertia. It is customary to disregard the issue, especially in
elementary discussions, and simply accept uncritically the Newtonian view
that there is such a thing as absolute acceleration (and we know it when we
see it), independent of the mean state of motion of all the matter in the
universe. Only on this naïve basis can we assert that inertial forces are “fictitious”,
i.e., that they do not arise from interactions. General relativity clearly undermines
this distinction between real and fictitious forces, because it teaches us
that the metric field responsible for the “real” force of gravity is
identical with the metric field responsible for the “fictitious” force of
inertia.